Today was the culminating lesson of Open Up Resources 6-8 Math grade 8 unit 2, lesson 13. I wanted it to go better than the last lesson of Unit 1 went with students not really getting too far past the warm-up activity of exploring angles of pattern blocks.
This lesson started with a notice and wonder of 2 photos of 3 pens, one outdoors with equal shadows and the other indoors with a lamp as light source. Students were very intrigued.
Here are the images from reference, linked from the lesson plan freely available above:
After 2 minutes of quiet think time and passing out the lesson 12 cool down, I gave classes a minute to share with their partner before writing their thoughts on a T chart (MLR 2: Collect & Display). Here were highlights from 2 classes:
In the slide show the next photo draws lines from the light sources to show the angles created by the shadows indoors are not equal but the light rays from the sun outside create parallel light rays that make the angles with the ground equal.
Before jumping into the lamp post activity, I re engaged students with the lesson 12 cooldown that challenged students to check if a point was on the graph of a non proportional equation. When students took the cool-down, I prompted students who were stuck or blanking out to try to find the slope of the line. Before looking at each classes results, I worked the steps myself to guide my thinking of how students may approach the task and to plant a seed on how I wanted students to re-engage with the task the following day.
This is where I created some comment codes, that I’ve talked about in our weekly Open Up Resources 6-8 Math chat at 5:30 PST every Monday, and many people have inquired how I do it. Here is a perfect example to show.
Students received an “S” if they were unable to identify the slope of the line. If they did figure it out, no “S” code. If they weren’t able to set-up an expression or equation equal to the slope, they got an “E.” If they didn’t check if the point was on the line or incorrectly used their expression or equation (switching order of ratio, switching x and y coordinates) they got a “P.” So the less codes you got, the more you understood the task. My general marking codes are similar to the ones I learned in MARS tasks: check mark for correct, x for incorrect, and ^ (caret) for correct but incomplete thinking, that wouldn’t get full credit on an assessment.
To re-engage students the next day, I asked students if this was a proportional relationship. They said no, it doesn’t pass through the origin. Some kids divided the y coordinate by the x coordinate to see if they would equal to the slope, 1/2. They were referring to the classwork activity a day before where the line was y=3/4x.
So, I asked the class what the slope was. Many said, 2/4 or 1/2. I then asked raise your hand and prove it. So many talked about drawing a triangle between two points, and labeling the vertical and horizontal lengths, and writing vertical/horizontal.
I asked students to call out lattice points, points that were on the line and exactly on the corner. As they called them out, I marked and labeled them. I then labeled how the student volunteer found the slope, and pushed students to see how the slope was related to the pair of coordinates. Kids saw the vertical was the difference between the y values, and the horizontal was the difference of the x values.
Then I said, let’s mark the point at the edge of the graph, not a lattice point, that we aren’t sure of the coordinate and call it (x,y). This represents a point anywhere on the line. If we can make a slope triangle using this, we can find out if any point is on the line. Then I asked how to label the slope triangle between (x,y) and (8,7). [I was careful to not pick a point whose x and y coordinates were the same, as this would not help differentiate the x and the y coordinates]. They said y-7, then x-8. I asked students what we should do with those expressions, and they said divide the vertical by the horizontal. I replied, and that should equal to…? (1/2). I reminded them this is true because the slope is equal between any two triangles on the line because they are… similar! [I’ve noticed a lot of mix up between the words corresponding, congruent, and similar. It’s a lot of language demands for students]
Then I asked how do we see if (20,13) is on the line? I honored that some kids made a mistake here and I understood that for slope vertical is the difference in the y coordinates first, but in an ordered pair, y is not first, x is first. A few kids switched this around. I asked what should we write instead of y-7? (13-7!) Instead of x-8? (20-8). That equals 6/12, and wala, it is equivalent to the slope, 1/2. I also had students write a sentence, declaring that answer: “(20,13) is on the line because when you substitute 13 for y and 20 for x in the equation, it equals 1/2, the slope.”
I have to show what this student did. Although there’s no equation, the SMP’s are evident all over this with the perseverance, and the repeated reasoning of a pattern.
Then we jumped back into the lesson. Students had 2 minutes to notice any relationships between the heights of the people and lamppost and their shadows:
My students asked if I was the guy in the middle. I did admit that this gentleman is the same height as me (72 inches or 6 feet), although he has 2 sons, and I have 2 daughters much younger.
Most kids found the approximate constant of proportionality (1.5), but in my first class, one student had a REALLY clever way of finding the lamp post height. I mentioned him, Carson, in later classes to show students there was an alternative method. He got his protractor and measured the man’s shadow and got 2 centimeters. He then measured the lamp post shadow and got 5 centimeters. He reasoned that 2 times 2.5 is 5 centimeters. So, he said that the lamp post must be the same size of 2 and a half men! He said 72 plus 72 is 144, and half of 72 is 36, so 144+36 is 180, which is pretty darn close to the answer students got of 171 inches when multiplying the lamp posts’ shadow by 1.5.
The next activity is pure genius. After that activity synthesis, it fades out the photo and draws black lines on the angles created by the people and their shadows. Students are given 2 minutes to come up with their first rough draft explanation of why there is an approximate proportional relationship between shadows and heights.
This was my first time I was going to use the MLR 1 (Math Language Routine), Stronger and Clearer Each Time with successive pair shares. I first experienced this routine at Asilomar last year in Chrissy Newell’s session that was right after my session. I didn’t follow the instructions exactly how I was supposed to, after referring to the Course Guide outlining the procedure, I realize that students were supposed to think about what they were going to say to their partner without looking at what they first wrote. I will read through that again before I try this great routine again.
When I executed the routine, having cards taped to each desk for random seating really helped. Students matched up with students at the table across from them with the same suit to share their thoughts, listen to their partners, and critique each others reasoning and add to what they wrote (some had wrote nothing in their independent time). I could see students who usually didn’t produce as much work output, working harder to incorporate notes from their partner’s ideas. I then had them think and write a bit, then pair up with a different table. This really helped in the activity synthesis because students were more willing to share one idea they had.
The main point to drive home was that the shadows and objects created triangles that were SIMILAR. Yes, they were all right triangles, but that only proves that which angles were the same? The 90 degree one. So, students, how many pairs of angles must be the same to be similar? (2!) So, how about a second pair? They saw that they were corresponding because the light rays appear to be all be parallel. This means that the side lengths are proportional to each other, and are scaled copies.
Then students went outside with clear roles: 1 student would stand, whoever knew their height by heart. Two students would measure with a meter/yard stick, and the 4th group member recorded the data. Then they measured the shadow of a tree or a tall object and then went in to do their calculations. We ran out of time, and one of my classes had inattentive behavior that slowed the flow of the class period which made me omit the outside portion and just have a longer conversation about why the relationship was proportional.
I am really happy with how this lesson went. With this curriculum, Brooke Powers reminded me that my students are making memories of learning experiences in math class that I believe they will remember and recall later on.
As Illustrative Mathematics says, the units and lessons follow this model:
I really am happy with how students consolidated and applied their learning in this culminating unit 2 lesson. I am also excited to try successive pair shares (MLR 1) again with a different concept.